On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley Torvik equation S. Saha Ray Department of Mathematics, National Institute of Technology, Orissa, Rourkela 769 008, India article info Keywords: Fractional differential equation Bagley Torvik equation Haar wavelets Operational matrices abstract In the present analysis, the motion of an immersed plate in a Newtonian fluid described by Torvik and Bagley’s fractional differential equation [1] has been considered. This Bagley Torvik equation has been solved by operational matrix of Haar wavelet method. The obtained result is compared with analytical solution suggested by Podlubny [2]. Haar wavelet method is used because its computation is simple as it converts the problem into algebraic matrix equation. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction Fractional calculus is a field of applied mathematics that deals with derivatives and integrals of arbitrary orders (including complex orders). It is also known as generalized integral and differential calculus of arbitrary order Kilbas et al. [3] and Saba- tier et al. [4]. Fractional calculus was described by Gorenflo and Mainardi [5] as the field of mathematical analysis which deals with investigation and applications of integrals and derivatives of arbitrary order. Fractional calculus is in use for past 300 years ago. And many great mathematician (pure and applied) Sabatier et al. [4] such as Abel, Caputo, Euler, Fourier, GrÜnwald, Hadamard, Hardy, Heaviside, Holmgren, Laplace, Leibniz, Letnikov, Lioville, Riemann, Riesz and Weyl made major contributions to the theory of fractional calculus. The history of fractional calculus was started at the end of the 17th century and the birth of fractional calculus was due to a letter exchange. At that time scientific journals did not exist and Scientist were exchange their information through letters. The first conference on fractional calculus and its applications was organized in June 1974 by Ross and held at university of new Haven. In recent years, fractional calculus has become the focus of interest for many researchers in different disciplines of applied science and engineering because of the fact that a realistic modeling of a physical phenomenon can be successfully achieved by using fractional calculus. The fractional derivative has been occurring in many physical problems such as frequency dependent damping behavior of materials, motion of a large thin plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, the PI k D l controller for the control of dynamical systems etc. Phenomena in electromagnetics, acoustics, viscoelasticity, and elec- trochemistry and material science are also described by differential equations of fractional order. The solution of the differ- ential equation containing fractional derivative is much involved. Fractional calculus has been used to model physical and engineering processes that are found to be best described by frac- tional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differ- ential equations. 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.11.007 E-mail addresses: santanusaharay@yahoo.com, saharays@nitrkl.ac.in Applied Mathematics and Computation 218 (2012) 5239–5248 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc